Topological zero modes and Dirac points protected by spatial symmetry and chiral symmetry

We explore a new class of topologically stable zero-energy modes which are protected by coexisting chiral and spatial symmetries. If a chiral-symmetric Hamiltonian has an additional spatial symmetry such as reflection, inversion, and rotation, the Hamiltonian can be separated into independent chiral-symmetric subsystems by the eigenvalue of the space symmetry operator. Each subsystem supports chiral zero-energy modes when a topological index assigned to the block is nonzero. By applying the argument to Bloch electron systems, we detect band touching at symmetric points in the Brillouin zone. In particular, we show that Weyl nodes appearing in honeycomb lattice (e. g., graphene) and in half-flux square lattice are protected by threefold and twofold rotation symmetry, respectively. We also present several examples of Dirac semimetal with isolated band-touching points in three-dimensional k space, which are protected by combined symmetry of rotation and reflection. The zero-mode protection by spatial symmetry is distinct from that by the conventional winding number. We demonstrate that symmetry-protected band touching points emerge even though the winding number is zero. Finally, we identify relevant topological charges assigned to the gapless points.